Symmetric polynomials are interesting not just in their own right. They appear naturally in various contexts, e.g., as characters in representation theory and as cohomology classes of Grassmannians and other homogeneous spaces in geometry. There are many different bases for the space of symmetric polynomials, perhaps the most interesting of which is the one formed by Schur polynomials. The Littlewood-Richardson (LR) rule expresses the product of two given Schur polynomials as a linear combination of Schur polynomials. The first half of the talk will be a tour of these topics that should be accessible even to undergraduate students.
The second half will be an exposition (which hopefully will continue to be widely accessible!) of recent joint research work with Mrigendra Singh Kushwaha and Sankaran Viswanath, both of IMSc. The LR rule above has a natural interpretation as giving the decomposition as a direct sum of irreducibles of the tensor product of two irreducible representations of the unitary (or general linear) group. A generalised version of it (due to Littelmann, still called the LR rule) gives the analogous decomposition for any reductive group. On the tensor product of two irreducible representations, there is the natural “Kostant-Kumar” filtration indexed by the Weyl group. This consists of the cyclic submodules generated by the highest weight vector tensor an extremal weight vector. We obtain a refined LR rule that gives the decomposition as a direct sum of irreducibles of the Kostant-Kumar submodules (of the tensor product). As an application, we obtain alternative proofs of refinements of “PRV type” results proved by Kumar and others. (PRV = Parthasarathy, Ranga Rao, Varadarajan)